EasyChair Preprint № 5663 Generating Custom Set Theories with Non-Set Structured Objects Ciarán Dunne, Joe Wells and Fairouz Kamareddine EasyChair preprints are intended for rapid dissemination of research results and are integrated with the rest of EasyChair. June 1, 2021 Generating Custom Set Theories with Non-Set Structured Objects Ciar´anDunne, J. B. Wells, Fairouz Kamareddine Heriot-Watt University Abstract. Set theory has long been viewed as a foundation of math- ematics, is pervasive in mathematical culture, and is explicitly used by much written mathematics. Because arrangements of sets can represent a vast multitude of mathematical objects, in most set theories every object is a set. This causes confusion and adds difficulties to formalising math- ematics in set theory. We wish to have set theory's features while also having many mathematical objects not be sets. A generalized set theory (GST) is a theory that has pure sets and may also have non-sets that can have internal structure and impure sets that mix sets and non-sets. This paper provides a GST-building framework. We show example GSTs that have sets and also (1) non-set ordered pairs, (2) non-set natural numbers, (3) a non-set exception object that can not be inside another object, and (4) modular combinations of these features. We show how to axiomatize GSTs and how to build models for GSTs in other GSTs. 1 Introduction Set Theory as a Foundation of Mathematics. Set theories like Zermelo- Fraenkel (ZF), and closely related set theories like ZFC and Tarski-Grothendieck (TG), play many important roles in mathematics. ZF's axioms allow express- ing a vast number of mathematical concepts. For most of the last century most mathematicians have accepted theories like ZF as suitable foundations of math- ematics. ZF's axioms have been rigorously evaluated for roughly a century and have no known inconsistencies. Mathematical theories are often assessed against the standard of whether models can be constructed for them in theories like ZF (what Maddy [14] calls risk assessment). Much of mathematical notation and reasoning is rooted in set theory. A significant amount of mathematics has been formalised in set theory and computer-verified using proof assistants like Isabelle/ZF [18,9], Mizar [2], and Metamath [15]. Mathematics varies in the kind and degree of assumptions made of the under- lying foundation. Some mathematics explicitly specifies a set-theoretic or type- theoretic foundation and some does not. Set theories like ZF are usually stated in first-order logic (FOL), but are sometimes stated in higher-order logic (HOL) or given as theories embedded in a dependent type system. In some mathematics, functions are sets of ordered pairs while in other mathematics functions are not even sets. There is variation in how undefined terms are treated [6,19]. When 2 C. Dunne, J. B. Wells, F. Kamareddine viewing ZF as the underlying foundation, it is assumed that high-level mathe- matics has meaningful translations into ZF, or that ZF can be safely modified to accommodate the user's needs. Representation Overlap in Set-Theoretic Formalisation. Translating human-written mathematics into ZF has complications. Every object in ZF's domain of discourse is a pure set, so objects of human-written text must be represented as pure sets. Objects that the mathematician views as distinct can have the same ZF representation. For example, consider formalising a function g :(N2[P(N)) ! f0; 1g such that g(h0; 1i) = 0 and g(f1; 2g) = 1. Let ( · )∗ be the translation of human-written mathematical objects into the domain of ZF. Typi- cally, N is represented using the von Neumann ordinals, so 0∗ := ; and (k+1)∗ := k∗ [ fk∗g. Also, ordered pairs are usually represented using Kuratowski's en- coding where ha; bi∗ := ffa∗g; fa∗; b∗gg. Furthermore, sets of the human-written ∗ ∗ ∗ text usually get the na¨ıve translation fx1; : : : ; xng = fx1; : : : ; xng. Using these representations, the ordered pair h0; 1i and the set f1; 2g are represented in ZF by the same pure set: ff;g; f;; f;ggg. Thus, a na¨ıve translation of the definition of g will require that 0∗ = g(h0; 1i∗) = g(f1; 2g∗) = 1∗ and there will be no value for g satisfying its specification, i.e., the na¨ıvely translated definition will fail to define anything. A standard set-theoretic solution is to not use N2 [P(N) as the domain of g but instead to use (f0g × N2) [ (f1g × P(N)), i.e., tag every member of N2 with 0 and every member of P(N) with 1. This works because f0g × N2 and f1g × P(N) are disjoint. This requires complete foresight of the objects to be used, obscures the mathematics under a layer of tagging and untagging, and increases the costs of formalisation. Furthermore, sometimes the mathematics needs a class (sometimes a proper class) of objects that are distinct from all sets, adding complication. And some- times the objects that must be distinct from all sets can contain sets. The proper set-theoretic solution is to build a hierarchy in ZF to represent both the sets and the non-set objects of the human-written mathematics using a construction sim- ilar to the von Neumann cumulative hierarchy. An example of doing this is the set theory ZFP [5] which has proper classes of both sets and non-set ordered pairs. A model of ZFP can be built in ZF using tagged ZF sets to represent both ZFP sets and ZFP non-set ordered pairs. Proper classes and tagging both involve awkward reasoning. Definitions, lem- mas, and proofs quickly become messy. The user must redefine and reprove op- erations and relations, leaving them with duplicate symbols and concepts (e.g., the power set operation of ZF vs. the analogous operation on the tagged sets within ZF that represent the sets of the human-written mathematics). How to build these models is not obvious to most mathematicians. Type Theory as an Alternative. Type theories typically avoid representation overlap by preventing operations that mix types. Operating on multiple types is done via sum types or inductive datatypes, and something equivalent to tagging Generating Custom Set Theories with Non-Set Structured Objects 3 and untagging happens in a type theory's underlying model theory, but the user is shielded from most details. Unfortunately, formalising mathematics in type theory is not always the best option. Removing set-theoretic dependencies can transform a text in ways that take it far from the author's conception. As mathematics gets more complex, the typing combinations push the limits of human cognition. Type error mes- sages can be beyond human comprehension. The typing rules of proof assistants can differ in significant ways from the human-readable documentation, and im- mense expertise in the implementation can be needed. Typing constraints can add proving obligations that are not relevant to the mathematics being for- malised. Formalising mathematics in type theories can require awkward and expensive workarounds that sometimes seem infeasible. To address these issues, more sophisticated type systems are developed that can require more expertise to comprehend. Finally, some type-theoretic provers focus on constructive, non- classical reasoning, but much mathematics is non-constructive, and constructive reasoning can be an unnecessary burden. An Arena for Custom Set Theories. We seek to retain the useful qualities of set theory whilst being able to have mathematical objects that are genuinely not sets. Some set theories (e.g., ZFA, KPU) have non-set objects called urelements which contain no set members (but are not the empty set) and can belong to sets. Typically urelements have no internal structure (an exception is Aczel's GST [1]). To avoid confusion with typical structure-free urelements, we use the phrase non-set object for members of a domain that are not sets. A set is pure iff all of its members are pure sets; other sets are impure.A generalized set theory (GST) is a theory that has pure sets and may also have non-sets that can have internal structure and impure sets. ZFP (mentioned above) is a GST with non-sets with internal structure. If a set theory S can be shown to be consistent relative to ZF and S is a better match for some mathematicians' needs, it is reasonable that they use S instead of ZF as a foundation. So we ask: Is it possible to give each mathematician a foundation that matches their intuition and in which their mathematics is formally true in the original human-written form rather than only becoming formally true after substantial effort and transformation? With this aim in mind, we propose what we call an arena in which multiple different GSTs (including ZF) can co-exist and a toolkit to support showing relative consistency results. Our plan and its fulfillment in later sections of this paper is as follows. We begin in section 2 with a logical framework that supplies features needed for an arena for set theories. Our design is inspired by systems such as the com- bination of Isabelle/Pure with Isabelle/FOL that underlies Isabelle/ZF, but we have deliberately used a bare minimum of features. To cleanly support multiple GSTs simultaneously, we have a countably infinite set of domain types which are base types of individuals. We have function types to support definitions and set theory axiom schemas. In any given derivation, one domain type is designated as the founder domain type. We allow 8-introduction only at the founder domain 4 C. Dunne, J. B. Wells, F. Kamareddine type. We allow 8-elimination at the founder domain type and all non-domain types, and forbid 8-elimination at all non-founder domain types.